Question

At a Sci-fi convention, the contestants for the costume contest had 3 different themed costumed categories. There were 12 who wore Space themed costumes (Star wars, Babylon 5, FireFly, Star Trek, etc.), 18 who were Video game themed costumes (Assassin's Creed, Zelda, Metal Gear, Final Fantasy, etc.), and 10 who wore Alternative Reality themed costumes (steampunk, magic based, werewolves, etc.).

a) How many ways can you introduce all of the Space themed contestants

b) This contest allows the audience to vote on their top 3, by putting a 1, 2 and 3 next to the name of the contestant's name. (Statement for question 1b)

1b) How many ways can the audience vote for their top 3.

c) What if all were judge separately and only the top 3 of each theme advanced to the next level, all winning $200, and a chance to compete for the grand prize of an additional $1000 (Statement for the 1c-4c questions)

1c) How many ways can they pick the top 3 in space themed costumes.

2c) How many ways can they pick the top 3 in Video game themed costumes

3c) How many ways can they pick the top 3 Alternative Reality themed costumes

4c) How many ways can they pick the grand prize winner from the top 3 candidates from Space, Video game and Alternative Reality Winners.

Answer 1

We have ,

Number of contestants who wore Space themed costumes = 12

Number of contestants who wore Video game themed costumes = 18

Number of contestants who wore Alternative Reality themed costumes =10

a) Number of ways all of the Space themed contestants can be introduced = 12 ! =479001600 ways

b) Total contestants = 12 + 18 + 10 = 40

So, number of ways the audience vote for their top 3 = ⁴⁰C₃ = 9880 ways

c)

1. Number of ways the top 3 in space themed costumes can be picked = ¹²C₃ = 220
2. Number of ways the top 3 in space themed costumes can be picked = ¹⁸C₃ = 816
3. Number of ways the top 3 in space themed costumes can be picked = ¹⁰C₃ = 120